Trigonometric Functions

Given $\cos x = -\frac{1}{2}$.

Using $\sin^2 x + \cos^2 x = 1$:

$\sin^2 x = 1 – (-\frac{1}{2})^2 = 1 – \frac{1}{4} = \frac{3}{4}$

$\sin x = \pm \frac{\sqrt{3}}{2}$. Since x is in Q3, $\sin x$ is negative.

Given $\sin x = \frac{3}{5}$.

Using $\sin^2 x + \cos^2 x = 1$:

$\cos^2 x = 1 – (\frac{3}{5})^2 = 1 – \frac{9}{25} = \frac{16}{25}$

$\cos x = \pm \frac{4}{5}$. Since x is in Q2, $\cos x$ is negative.

Given $\cot x = \frac{3}{4} \Rightarrow \tan x = \frac{4}{3}$.

Using $\sec^2 x = 1 + \tan^2 x$:

$\sec^2 x = 1 + (\frac{4}{3})^2 = 1 + \frac{16}{9} = \frac{25}{9}$

$\sec x = \pm \frac{5}{3}$. Since x is in Q3, $\sec x$ is negative.

Given $\sec x = \frac{13}{5} \Rightarrow \cos x = \frac{5}{13}$.

Using $\sin^2 x + \cos^2 x = 1$:

$\sin^2 x = 1 – (\frac{5}{13})^2 = \frac{144}{169}$

$\sin x = \pm \frac{12}{13}$. Since x is in Q4, $\sin x$ is negative.

Given $\tan x = -\frac{5}{12} \Rightarrow \cot x = -\frac{12}{5}$.

Using $\sec^2 x = 1 + \tan^2 x$:

$\sec^2 x = 1 + (-\frac{5}{12})^2 = \frac{169}{144}$

$\sec x = \pm \frac{13}{12}$. Since x is in Q2, $\sec x$ is negative.

Logic: $\sin(2n\pi + \theta) = \sin \theta$ or $\sin(n \times 360^\circ + \theta) = \sin \theta$.

Divide 765 by 360:

$765^\circ = 2 \times 360^\circ + 45^\circ$

Therefore, $\sin 765^\circ = \sin 45^\circ$

Logic: $\text{cosec}(-\theta) = -\text{cosec } \theta$.

$-\text{cosec}(1410^\circ)$. Divide 1410 by 360:

$1410^\circ = 4 \times 360^\circ – 30^\circ$ (Using nearest multiple)

$\text{cosec}(4 \times 360^\circ – 30^\circ) = \text{cosec}(-30^\circ) = -\text{cosec } 30^\circ$.

So, $-\text{cosec}(1410^\circ) = -(-\text{cosec } 30^\circ) = \text{cosec } 30^\circ$.

Logic: $\tan(n\pi + \theta) = \tan \theta$.

$\frac{19\pi}{3} = 6\pi + \frac{\pi}{3}$.

Since $6\pi$ is a multiple of $\pi$, we are in the 1st quadrant cycle.

$\tan(6\pi + \frac{\pi}{3}) = \tan \frac{\pi}{3} = \tan 60^\circ$

Logic: $\sin(-\theta) = -\sin \theta$.

$-\sin(\frac{11\pi}{3})$. Break down $\frac{11\pi}{3}$:

$\frac{11\pi}{3} = 4\pi – \frac{\pi}{3}$.

$\sin(4\pi – \frac{\pi}{3}) = \sin(-\frac{\pi}{3}) = -\sin \frac{\pi}{3}$.

Total expression: $-(-\sin \frac{\pi}{3}) = \sin \frac{\pi}{3}$.

Logic: $\cot(-\theta) = -\cot \theta$.

$-\cot(\frac{15\pi}{4})$. Break down $\frac{15\pi}{4}$:

$\frac{15\pi}{4} = 4\pi – \frac{\pi}{4}$.

$\cot(4\pi – \frac{\pi}{4}) = \cot(-\frac{\pi}{4}) = -\cot \frac{\pi}{4}$.

Total expression: $-(-\cot \frac{\pi}{4}) = \cot \frac{\pi}{4}$.